Research Interests



Algebraic Graph Theory

A classic question of plane geometry asks for the greatest area that can be enclosed by a figure of a given perimeter. This is known as the isoperimetric problem, and it has a pedigree going back to the ancient Greeks. A more modern formulation begins with an n-dimensional manifold and considers the set of all (n-1)-dimensional hypersurfaces that divide the manifold into two pieces. For each such hypersurface we compute the ratio of the area of the surface to the volume of the smaller of the two pieces, and we seek the the infimum of the set of such ratios. In 1970 Jeff Cheeger proved that the first eigenvalue of the Laplacian of the manifold can be bounded below in terms of this infimum, now called the Cheeger constant in his honor.

In 1982 Peter Buser established that the eigenvalue spectrum of the manifold could also be bounded above in terms of the Cheeger constant. Buser also introduced a combinatorial analog of the Cheeger constant that applies to finite, simple graphs. This analog is called the isoperimetric number and is defined as follows: We take a finite simple graph and consider its edge-cutsets. We then seek the cutset that minimizes the ratio of the number of edges in the set to the number of vertices in the smaller of the two pieces. This minimal ratio is the isoperimetric number of the graph. Since a graph is a sort of degenerate manifold, it is not surprising that the isoperimetric number is related to the eigenvalues of the graph.

Buser developed the isoperimetric number as a tool for studying spectral geometry; the idea was to gain information regarding the spectrum of a manifold based on properties of certain associated graphs. However, his creation is now of independent interest to combinatorialists. My own research to date has focussed on the isoperimetric properties of certain Cayley graphs arising from arithmetic Riemann surfaces. These surfaces arise from a consideration of the action of the modular group (or its congruence subgroups), via fractional linear transformations, on the complex upper half plane. I am also interested in how group-theoretic properties can be used to gain information regarding the isoperimetric properties of associated Cayley graphs.

Publications



Analytic Number Theory

The study of exponential sums has a long history in number theory. Gauss used them to prove the law of quadratic reciprocity, and his techniques were later used by other mathematicians to prove higher order reciprocity laws. The circle method of Hardy, Littlewood and Ramanujan leads naturally to a consideration of exponential sums when applied to Waring's problem. These are just two examples, among many, of the importance of exponential sums.

By an exponential sum we mean a finite sum whose summands are all found on the complex unit circle. The goal is to find upper and lower bounds on these sums, a task to which mathematicians have applied considerable intellectual firepower over the years. Keeping up with the state of the art can be a daunting task, as ever more sophisticated techniques are invoked to make ever more marginal improvements in existing bounds. Nonetheless, the subject is fascinating and worth the effort. My research in this area, conducted with Todd Cochrane and Christopher Pinner, has involved the mod p Waring's problem, as well as exponential sums whose exponents are given by sparse polynomials.

Publications



Evolutionary Biology

Charles Darwin's two major works on evolution, On the Origin of Species and The Descent of Man, contain more than a thousand pages of scientific brilliance, but not even a single equation. This is easy to understand, given Darwin's well-known antipathy toward mathematics. Indeed, since biology prior to Darwin was little more than a compendium of isolated facts about the natural world, it is unsurprising that no one had thought to look for mathematical regularities in the data.  That all changed in the first few decades of the twentieth century. The rediscovery of Mendelian genetics, coupled with advances in the understanding of the physical basis of heredity, led many to realize that asessing the viability of natural selection as a mechanism of evolution was partly a mathemtaics problem. Beginning with the efforts of the biometricians, who brought statistical techniques to bear on biological problems, and continuing through the more sophisticated work of Fisher, Haldane, and Wright, it was found that the short term flow of genetic information in wild populations could be mathematically modelled.

Mathematics continues to play an integral role in modern evolutionary theory. The game theoretic techniques pioneered by John Nash are now among the standard tools of ethology (the study of animal behavior). Paleontologists use probabilistic models to ferret out the genuine patterns in their data from the random noise. Geneticists make use of information theory in studying the genomes of organisms.

The use of mathematics in evolutionary theory raises a host of fascinating philosophical and practical questions. Excessive mathematical modelling is the trademark of reductionism, which many biologists find dissatisfying. Living creatures are far too complex, it is argued, to be represented as variables in equations. To what extent is this criticism valid? Given certain environmental initial conditions, how constrained is natural selection in crafting novel phenotypes? Is the distinction between micro and macro evolution genuine, and does mathematics have any light to shed on this question? It is my opinion that these are investigations more mathematicians should take an interest in.

Of course, evolutionary biology is interesting also for its major impact on culture. Religious critics of evolutionary theory, be they Young-Earth Creationists or the more sophisticated Intelligent-Design Theorists, routinely use mathematical arguments in their writings. This creates the illusion of scientific legitimacy; an illusion they peddle shamelessly to state school boards and legislatures, and even to Congress. The use of pseudomathematics as part of an attempt to influence public policy in ways that are potentially damaging to American science is something more mathematicians need to take a stand against.

Publications